Posts tagged differential equations
Exponential equations to model population growth

The population of a species that grows exponentially over time can be modeled by P(t)=Pe^(kt), where P(t) is the population after time t, P is the original population when t=0, and k is the growth constant.

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Solving initial value problems using laplace transforms

To use a Laplace transform to solve a second-order nonhomogeneous differential equations initial value problem, we’ll need to use a table of Laplace transforms or the definition of the Laplace transform to put the differential equation in terms of Y(s). Once we solve the resulting equation for Y(s), we’ll want to simplify it until we recognize that the terms in our equation match formulas in a table of Laplace transforms.

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Solving second-order homogeneous differential equations

The first thing we want to learn about second-order homogeneous differential equations is how to find their general solutions. The formula we’ll use for the general solution will depend on the kinds of roots we find for the differential equation.

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How to find the orthogonal trajectories

The orthogonal trajectories to a family of curves are the curves that intersect each member of the family at a perfectly perpendicular angle. So given a family of curves, you can change the value of the constant in the equation that models the family, to create a family of many curves, and then sketch the family in the plane.

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